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LIST OF FORMULAE by vfd10970

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MATHEMATICS

LIST OF FORMULAE
AND

STATISTICAL TABLES

(List MF1)
Pure Mathematics

Mensuration

Surface area of sphere = 4π r2
Area of curved surface of cone = π r × slant height

Trigonometry

a2 = b2 + c2 − 2bc cos A

Arithmetic Series

un = a + (n − 1)d
Sn = 1 n(a + l) = 1 n{2a + (n − 1)d}
2            2

Geometric Series

un = arn−1
a(1 − rn )
Sn =
1−r
a
S∞ =          for | r | < 1
1−r

Summations
n
∑ r2 = 1 n(n + 1)(2n + 1)
6
r=1
n
∑ r3 = 1 n2 (n + 1)2
4
r=1

Binomial Series

n        n          n+1
+           =
r       r+1         r+1
n          n                       n
(a + b)n = an +        an−1 b +     an−2 b2 + . . . +        an−r br + . . . + bn        n∈ ,
1          2                       r
n                n!
where        = n Cr =
r            r!(n − r)!
n(n − 1) 2           n(n − 1) . . . (n − r + 1) r
(1 + x)n = 1 + nx +          x + ... +                               x + ...      | x | < 1, n ∈
1.2                      1.2.3 . . . r

Logarithms and exponentials

ex ln a = ax

Complex Numbers

{r (cos θ + i sin θ )}n = r n (cos nθ + i sin nθ )
eiθ = cos θ + i sin θ
2π k i
The roots of    n
= 1 are given by    =e    n       , for k = 0, 1, 2, . . . , n − 1

2
Maclaurin’s Series
x2                   xr
f(x) = f(0) + xf (0) +     f (0) + . . . + f (r) (0) + . . .
2!                   r!
x2            xr
ex = exp(x) = 1 + x +      + ... +       + . . . for all x
2!            r!
x2 x3                        xr
ln(1 + x) = x −    +      − . . . + (−1)r+1 + . . . (−1 < x ≤ 1)
2    3                       r
x3 x5                       x2r+1
sin x = x −    +    − . . . + (−1)r              + . . . for all x
3! 5!                    (2r + 1)!
x2 x4                  x2r
cos x = 1 −      +   − . . . + (−1)r        + ...    for all x
2! 4!                 (2r )!
x3 x5                    x2r+1
tan−1 x = x −    +    − . . . + (−1)r         + . . . (−1 ≤ x ≤ 1)
3    5                  2r + 1
x3 x5               x2r+1
sinh x = x +    +    + ... +              + . . . for all x
3! 5!             (2r + 1)!
x2 x4          x2r
cosh x = 1 +      +   + ... +        + ...      for all x
2! 4!         (2r )!
x3 x5          x2r+1
tanh−1 x = x +      +   + ... +        + ...        (−1 < x < 1)
3   5         2r + 1

Hyperbolic Functions

cosh2 x − sinh2 x = 1
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh2 x + sinh2 x
√
cosh−1 x = ln{x + (x2 − 1)}       (x ≥ 1)
√
sinh−1 x = ln{x + (x2 + 1)}
1+x
tanh−1 x = 1 ln               |x| < 1
2       1−x

Coordinate Geometry
| ah + bk + c |
The perpendicular distance from (h, k) to ax + by + c = 0 is √ 2
(a + b2 )
m1 − m2
The acute angle between lines with gradients m1 and m2 is tan−1
1 + m1 m2

Trigonometric Identities

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan A ± tan B
tan(A ± B) =                        A±B≠ k+ 1 π
1 ∓ tan A tan B                    2

2t             1−t  2
For t = tan 1 A: sin A =          , cos A =
2             1+t   2           1 + t2
A+B           A−B
sin A + sin B = 2 sin         cos
2            2
A+B          A−B
sin A − sin B = 2 cos          sin
2            2
A+B           A−B
cos A + cos B = 2 cos           cos
2           2
A+B         A−B
cos A − cos B = −2 sin           sin
2           2
3
Vectors

a.b
The resolved part of a in the direction of b is
|b|
µa + λ b
The point dividing AB in the ratio λ : µ is
λ +µ
i a1 b1          a2 b3 − a3 b2
Vector product: a × b = | a | | b | sin θ n = j a2 b2 = a3 b1 − a1 b3
ˆ
k a3 b3          a1 b2 − a2 b1
If A is the point with position vector a = a1 i + a2 j + a3 k and the direction vector b is given by
b = b1 i + b2 j + b3 k, then the straight line through A with direction vector b has cartesian equation
x − a1 y − a2            − a3
=          =          (= λ )
b1         b2          b3
The plane through A with normal vector n = n1 i + n2 j + n3 k has cartesian equation
n1 x + n2 y + n3 + d = 0, where d = −a.n
The plane through non-collinear points A, B and C has vector equation
r = a + λ (b − a) + µ (c − a) = (1 − λ − µ )a + λ b + µ c
The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc
n1 α + n2 β + n3 γ + d
The perpendicular distance of (α , β , γ ) from n1 x + n2 y + n3 + d = 0 is      √ 2
(n1 + n2 + n2 )
2     3

Matrix transformations

cos θ       − sin θ
Anticlockwise rotation through θ about O:
sin θ        cos θ
cos 2θ       sin 2θ
Reﬂection in the line y = (tan θ )x:
sin 2θ     − cos 2θ

Differentiation

f(x)            f (x)

tan kx         k sec2 kx
1
sin−1 x        √
(1 − x2 )
1
cos−1 x       −√
(1 − x2 )
1
tan−1 x
1 + x2
sec x        sec x tan x
cot x        − cosec2 x
cosec x      − cosec x cot x
sinh x          cosh x
cosh x            sinh x
tanh x          sech2 x
1
sinh−1 x       √
(1 + x2 )
1
cosh−1 x       √ 2
(x − 1)
1
tanh−1 x
1 − x2

4
Integration ( + constant; a > 0 where relevant)

f(x)                                    f(x) dx

1
sec2 kx                                      tan kx
k
tan x                                    ln |sec x |
cot x                                    ln |sin x |
cosec x                      − ln |cosec x + cot x | = ln tan 1 x
2
sec x                  ln |sec x + tan x | = ln tan( 1 x + 1 π )
2     4
sinh x                                   cosh x
cosh x                                   sinh x
tanh x                                 ln cosh x
1                                −1 x
√ 2                               sin            |x| < a
(a − x2 )                                a
1                                  1         x
2 + x2
tan−1
a                                      a         a
1                    −1 x                 √ 2
√ 2                  cosh          or ln{x + (x − a2 )} (x > a)
(x − a2 )                     a
1                              x               √
√ 2                        sinh−1       or ln{x + (x2 + a2 )}
(a + x2 )                         a
1                  1       a+x       1          x
ln         = tanh−1              |x| < a
a 2 − x2              2a       a−x       a          a
1                                  1      x−a
ln
x 2 − a2                              2a      x+a
dv                       du
u      dx = uv −         v      dx
dx                       dx

Area of a sector

A=     1
2
r2 dθ    (polar coordinates)

dy    dx
A=     1
2
x      −y    dt          (parametric form)
dt    dt

Numerical Mathematics

Numerical integration

b
b−a
The trapezium rule:                 y dx ≈ 1 h{(y0 + yn ) + 2(y1 + y2 + . . . + yn−1 )}, where h =
2
a                                                                        n
b
Simpson’s Rule:             y dx ≈ 1 h{(y0 + yn ) + 4(y1 + y3 + . . . + yn−1 ) + 2(y2 + y4 + . . . + yn−2 )},
3
a
b−a
where h =       and n is even
n

Numerical Solution of Equations
f(xn )
The Newton-Raphson iteration for solving f(x) = 0: xn+1 = xn −
f (xn )

5
Mechanics

Motion in a circle

˙
Transverse velocity: v = rθ
Transverse acceleration: v = r θ
˙     ¨

˙2       v2
Radial acceleration: − rθ = −
r

Centres of Mass (for uniform bodies)

2
Triangular lamina:   3
along median from vertex
Solid hemisphere, radius r : 3 r from centre
8

Hemispherical shell, radius r: 1 r from centre
2
r sin α
Circular arc, radius r, angle at centre 2α :          from centre
α
2r sin α
Sector of circle, radius r, angle at centre 2α :           from centre
3α
Solid cone or pyramid of height h: 1 h above the base on the line from centre of base to vertex
4

Conical shell of height h: 1 h above the base on the line from centre of base to vertex
3

Moments of Inertia (for uniform bodies of mass m)

Thin rod, length 2l, about perpendicular axis through centre: 1 ml2
3

Rectangular lamina about axis in plane bisecting edges of length 2l: 1 ml2
3

Thin rod, length 2l, about perpendicular axis through end: 4 ml2
3

Rectangular lamina about edge perpendicular to edges of length 2l: 4 ml2
3

Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre: 1 m(a2 + b2 )
3

2

Disc or solid cylinder of radius r about axis: 1 mr 2
2

4

5

3

Parallel axes theorem: IA = IG + m(AG)2
Perpendicular axes theorem: I = Ix + Iy (for a lamina in the x-y plane)

6
Probability & Statistics

Probability

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A ∩ B) = P(A)P(B | A)
P(B | A)P(A)
P(A | B) =
P(B | A)P(A) + P(B | A )P(A )
P(Aj )P(B | Aj )
Bayes’ Theorem: P(Aj | B) =
ΣP(Ai )P(B | Ai )

Discrete distributions

For a discrete random variable X taking values xi with probabilities pi
Expectation (mean): E(X ) = µ = Σ xi pi
Variance: Var(X ) = σ 2 = Σ(xi − µ )2 pi = Σ xi pi − µ 2
2

For a function g(X ): E(g(X )) = Σ g(xi )pi
The probability generating function of X is GX (t) = E(tX ), and
E(X ) = GX (1)
Var(X ) = GX (1) + GX (1) − {GX (1)}2
For Z = X + Y , where X and Y are independent: GZ (t) = GX (t)GY (t)

Standard discrete distributions

Distribution of X                       P(X = x)           Mean        Variance      P.G.F.
n
Binomial B(n, p)                       px (1 − p)n−x        np      np(1 − p)     (1 − p + pt)n
x
λx
Poisson Po(λ )                           e−λ                 λ            λ          eλ (t−1)
x!
1          1−p            pt
Geometric Geo(p) on 1, 2, …            p(1 − p)x−1
p           p2       1 − (1 − p)t

Continuous distributions

For a continuous random variable X having probability density function f
Expectation (mean): E(X ) = µ =       xf(x) dx

Variance: Var(X ) = σ 2 =     (x − µ )2 f(x) dx =    x2 f(x) dx − µ 2

For a function g(X ): E(g(X )) =     g(x)f(x) dx
x
Cumulative distribution function: F(x) = P(X ≤ x) =               f(t) dt
−∞

The moment generating function of X is MX (t) = E(etX ) and
E(X ) = MX (0)
E(X n ) = M(n) (0)
X
Var(X ) = MX (0) − {MX (0)}2
For Z = X + Y , where X and Y are independent: MZ (t) = MX (t)MY (t)

7
Standard continuous distributions
Distribution of X                              P.D.F.              Mean             Variance         M.G.F.
1                                                   ebt − eat
Uniform (Rectangular) on [a, b]                                   1
(a   + b)       1
(b     − a)2
b−a               2                12                (b − a)t
1                       1          λ
Exponential                                     λ e−λ x
λ                       λ2        λ −t
1             x−µ 2
e− 2                                               eµ t+ 2 σ
1                                                     1     2 2
Normal N(µ , σ 2 )                         √               σ         µ                       σ2                   t
σ (2π )

Expectation algebra

Covariance: Cov(X , Y ) = E (X − µX )(Y − µY ) = E(XY ) − µX µY
Var(aX ± bY ) = a2 Var(X ) + b2 Var(Y ) ± 2ab Cov(X , Y )
Cov(X , Y )
Product moment correlation coefﬁcient: ρ =
σX σY
If X = aX + b and Y = cY + d, then Cov(X , Y ) = ac Cov(X , Y )
For independent random variables X and Y
E(XY ) = E(X )E(Y )
Var(aX ± bY ) = a2 Var(X ) + b2 Var(Y )

Sampling distributions

For a random sample X1 , X2 , . . . , Xn of n independent observations from a distribution having mean µ
and variance σ 2
σ2
X is an unbiased estimator of µ , with Var(X ) =
n
Σ(Xi − X )2
S2 is an unbiased estimator of σ 2 , where S2 =
n−1
For a random sample of n observations from N(µ , σ 2 )
X−µ
√ ∼ N(0, 1)
σ/ n
X−µ
√ ∼ tn−1       (also valid in matched-pairs situations)
S/ n
If X is the observed number of successes in n independent Bernoulli trials in each of which the
X
probability of success is p, and Y = , then
n
p(1 − p)
E(Y ) = p and Var(Y ) =
n
For a random sample of nx observations from N(µx , σx ) and, independently, a random sample of
2

ny observations from N(µy , σy )
2

(X − Y ) − (µx − µy )
∼ N(0, 1)
σx σy
2   2
+
nx ny
(X − Y ) − (µx − µy )
If σx = σy = σ 2 (unknown) then
2    2
∼ tn   +ny −2
,
1   1                 x
2
Sp      +
nx ny

2
(nx − 1)S2 + (ny − 1)Sy
x
2
where Sp =
nx + ny − 2

8
Correlation and regression

For a set of n pairs of values (xi , yi )
(Σ xi )2
Sxx = Σ(xi − x) =2
Σ xi
2
−
n
(Σ yi )2
Syy = Σ(yi − y)2 = Σ y2 −
i            n
(Σ xi )(Σ yi )
Sxy = Σ(xi − x)(yi − y) = Σ xi yi −
n
The product moment correlation coefﬁcient is
(Σ xi )(Σ yi )
Sxy             Σ(xi − x)(yi − y)                                   Σ xi yi −
r= √           =                                            =                                  n
(Sxx Syy )          Σ(xi − x)2 Σ(yi − y)2                                  (Σ xi )2               (Σ yi )2
Σ xi −
2                   Σ y2 −
i
n                       n
6Σ d2
Spearman’s rank correlation coefﬁcient is rs = 1 −
n(n2 − 1)
Sxy        Σ(xi − x)(yi − y)
The regression coefﬁcient of y on x is b =                  =
Sxx               Σ(xi − x)2
Least squares regression line of y on x is y = a + bx where a = y − bx

Distribution-free (non-parametric) tests

(Oi − Ei )2
Goodness-of-ﬁt test and contingency tables:              ∑              Ei
∼ χν
2

Approximate distributions for large samples
Wilcoxon Signed Rank test: T ∼ N 1 n(n + 1), 24 n(n + 1)(2n + 1)
4
1

Wilcoxon Rank Sum test (samples of sizes m and n, with m ≤ n):
W ∼ N 1 m(m + n + 1),
2
1
12
mn(m      + n + 1)

9
CUMULATIVE BINOMIAL PROBABILITIES
n=5
p      0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0    0.7738   0.5905   0.4437   0.4019   0.3277   0.2373   0.1681   0.1317   0.1160   0.0778   0.0503   0.0313   0.0185   0.0102   0.0053   0.0041   0.0024   0.0010   0.0003   0.0001   0.0001   0.0000   0.0000
1   0.9774   0.9185   0.8352   0.8038   0.7373   0.6328   0.5282   0.4609   0.4284   0.3370   0.2562   0.1875   0.1312   0.0870   0.0540   0.0453   0.0308   0.0156   0.0067   0.0033   0.0022   0.0005   0.0000
2   0.9988   0.9914   0.9734   0.9645   0.9421   0.8965   0.8369   0.7901   0.7648   0.6826   0.5931   0.5000   0.4069   0.3174   0.2352   0.2099   0.1631   0.1035   0.0579   0.0355   0.0266   0.0086   0.0012
3   1.0000   0.9995   0.9978   0.9967   0.9933   0.9844   0.9692   0.9547   0.9460   0.9130   0.8688   0.8125   0.7438   0.6630   0.5716   0.5391   0.4718   0.3672   0.2627   0.1962   0.1648   0.0815   0.0226
4   1.0000   1.0000   0.9999   0.9999   0.9997   0.9990   0.9976   0.9959   0.9947   0.9898   0.9815   0.9688   0.9497   0.9222   0.8840   0.8683   0.8319   0.7627   0.6723   0.5981   0.5563   0.4095   0.2262
5   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

n=6
p      0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0    0.7351   0.5314   0.3771   0.3349   0.2621   0.1780   0.1176   0.0878   0.0754   0.0467   0.0277   0.0156   0.0083   0.0041   0.0018   0.0014   0.0007   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000
1   0.9672   0.8857   0.7765   0.7368   0.6554   0.5339   0.4202   0.3512   0.3191   0.2333   0.1636   0.1094   0.0692   0.0410   0.0223   0.0178   0.0109   0.0046   0.0016   0.0007   0.0004   0.0001   0.0000
2   0.9978   0.9842   0.9527   0.9377   0.9011   0.8306   0.7443   0.6804   0.6471   0.5443   0.4415   0.3438   0.2553   0.1792   0.1174   0.1001   0.0705   0.0376   0.0170   0.0087   0.0059   0.0013   0.0001
3   0.9999   0.9987   0.9941   0.9913   0.9830   0.9624   0.9295   0.8999   0.8826   0.8208   0.7447   0.6563   0.5585   0.4557   0.3529   0.3196   0.2557   0.1694   0.0989   0.0623   0.0473   0.0159   0.0022
4   1.0000   0.9999   0.9996   0.9993   0.9984   0.9954   0.9891   0.9822   0.9777   0.9590   0.9308   0.8906   0.8364   0.7667   0.6809   0.6488   0.5798   0.4661   0.3446   0.2632   0.2235   0.1143   0.0328
5   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9993   0.9986   0.9982   0.9959   0.9917   0.9844   0.9723   0.9533   0.9246   0.9122   0.8824   0.8220   0.7379   0.6651   0.6229   0.4686   0.2649
6   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

n=7
p      0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95

10
x=0    0.6983   0.4783   0.3206   0.2791   0.2097   0.1335   0.0824   0.0585   0.0490   0.0280   0.0152   0.0078   0.0037   0.0016   0.0006   0.0005   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000
1   0.9556   0.8503   0.7166   0.6698   0.5767   0.4449   0.3294   0.2634   0.2338   0.1586   0.1024   0.0625   0.0357   0.0188   0.0090   0.0069   0.0038   0.0013   0.0004   0.0001   0.0001   0.0000   0.0000
2   0.9962   0.9743   0.9262   0.9042   0.8520   0.7564   0.6471   0.5706   0.5323   0.4199   0.3164   0.2266   0.1529   0.0963   0.0556   0.0453   0.0288   0.0129   0.0047   0.0020   0.0012   0.0002   0.0000
3   0.9998   0.9973   0.9879   0.9824   0.9667   0.9294   0.8740   0.8267   0.8002   0.7102   0.6083   0.5000   0.3917   0.2898   0.1998   0.1733   0.1260   0.0706   0.0333   0.0176   0.0121   0.0027   0.0002
4   1.0000   0.9998   0.9988   0.9980   0.9953   0.9871   0.9712   0.9547   0.9444   0.9037   0.8471   0.7734   0.6836   0.5801   0.4677   0.4294   0.3529   0.2436   0.1480   0.0958   0.0738   0.0257   0.0038
5   1.0000   1.0000   0.9999   0.9999   0.9996   0.9987   0.9962   0.9931   0.9910   0.9812   0.9643   0.9375   0.8976   0.8414   0.7662   0.7366   0.6706   0.5551   0.4233   0.3302   0.2834   0.1497   0.0444
6   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9995   0.9994   0.9984   0.9963   0.9922   0.9848   0.9720   0.9510   0.9415   0.9176   0.8665   0.7903   0.7209   0.6794   0.5217   0.3017
7   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

n=8
p      0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0    0.6634   0.4305   0.2725   0.2326   0.1678   0.1001   0.0576   0.0390   0.0319   0.0168   0.0084   0.0039   0.0017   0.0007   0.0002   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1   0.9428   0.8131   0.6572   0.6047   0.5033   0.3671   0.2553   0.1951   0.1691   0.1064   0.0632   0.0352   0.0181   0.0085   0.0036   0.0026   0.0013   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000
2   0.9942   0.9619   0.8948   0.8652   0.7969   0.6785   0.5518   0.4682   0.4278   0.3154   0.2201   0.1445   0.0885   0.0498   0.0253   0.0197   0.0113   0.0042   0.0012   0.0004   0.0002   0.0000   0.0000
3   0.9996   0.9950   0.9786   0.9693   0.9437   0.8862   0.8059   0.7414   0.7064   0.5941   0.4770   0.3633   0.2604   0.1737   0.1061   0.0879   0.0580   0.0273   0.0104   0.0046   0.0029   0.0004   0.0000
4   1.0000   0.9996   0.9971   0.9954   0.9896   0.9727   0.9420   0.9121   0.8939   0.8263   0.7396   0.6367   0.5230   0.4059   0.2936   0.2586   0.1941   0.1138   0.0563   0.0307   0.0214   0.0050   0.0004
5   1.0000   1.0000   0.9998   0.9996   0.9988   0.9958   0.9887   0.9803   0.9747   0.9502   0.9115   0.8555   0.7799   0.6846   0.5722   0.5318   0.4482   0.3215   0.2031   0.1348   0.1052   0.0381   0.0058
6   1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9987   0.9974   0.9964   0.9915   0.9819   0.9648   0.9368   0.8936   0.8309   0.8049   0.7447   0.6329   0.4967   0.3953   0.3428   0.1869   0.0572
7   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9998   0.9993   0.9983   0.9961   0.9916   0.9832   0.9681   0.9610   0.9424   0.8999   0.8322   0.7674   0.7275   0.5695   0.3366
8   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000
CUMULATIVE BINOMIAL PROBABILITIES
n=9
p        0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0      0.6302   0.3874   0.2316   0.1938   0.1342   0.0751   0.0404   0.0260   0.0207   0.0101   0.0046   0.0020   0.0008   0.0003   0.0001   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1     0.9288   0.7748   0.5995   0.5427   0.4362   0.3003   0.1960   0.1431   0.1211   0.0705   0.0385   0.0195   0.0091   0.0038   0.0014   0.0010   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000
2     0.9916   0.9470   0.8591   0.8217   0.7382   0.6007   0.4628   0.3772   0.3373   0.2318   0.1495   0.0898   0.0498   0.0250   0.0112   0.0083   0.0043   0.0013   0.0003   0.0001   0.0000   0.0000   0.0000
3     0.9994   0.9917   0.9661   0.9520   0.9144   0.8343   0.7297   0.6503   0.6089   0.4826   0.3614   0.2539   0.1658   0.0994   0.0536   0.0424   0.0253   0.0100   0.0031   0.0011   0.0006   0.0001   0.0000
4     1.0000   0.9991   0.9944   0.9910   0.9804   0.9511   0.9012   0.8552   0.8283   0.7334   0.6214   0.5000   0.3786   0.2666   0.1717   0.1448   0.0988   0.0489   0.0196   0.0090   0.0056   0.0009   0.0000
5     1.0000   0.9999   0.9994   0.9989   0.9969   0.9900   0.9747   0.9576   0.9464   0.9006   0.8342   0.7461   0.6386   0.5174   0.3911   0.3497   0.2703   0.1657   0.0856   0.0480   0.0339   0.0083   0.0006
6     1.0000   1.0000   1.0000   0.9999   0.9997   0.9987   0.9957   0.9917   0.9888   0.9750   0.9502   0.9102   0.8505   0.7682   0.6627   0.6228   0.5372   0.3993   0.2618   0.1783   0.1409   0.0530   0.0084
7     1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9990   0.9986   0.9962   0.9909   0.9805   0.9615   0.9295   0.8789   0.8569   0.8040   0.6997   0.5638   0.4573   0.4005   0.2252   0.0712
8     1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9999   0.9997   0.9992   0.9980   0.9954   0.9899   0.9793   0.9740   0.9596   0.9249   0.8658   0.8062   0.7684   0.6126   0.3698
9     1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

n = 10
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0     0.5987   0.3487   0.1969   0.1615   0.1074   0.0563   0.0282   0.0173   0.0135   0.0060   0.0025   0.0010   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.9139   0.7361   0.5443   0.4845   0.3758   0.2440   0.1493   0.1040   0.0860   0.0464   0.0233   0.0107   0.0045   0.0017   0.0005   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.9885   0.9298   0.8202   0.7752   0.6778   0.5256   0.3828   0.2991   0.2616   0.1673   0.0996   0.0547   0.0274   0.0123   0.0048   0.0034   0.0016   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000
3    0.9990   0.9872   0.9500   0.9303   0.8791   0.7759   0.6496   0.5593   0.5138   0.3823   0.2660   0.1719   0.1020   0.0548   0.0260   0.0197   0.0106   0.0035   0.0009   0.0003   0.0001   0.0000   0.0000
4    0.9999   0.9984   0.9901   0.9845   0.9672   0.9219   0.8497   0.7869   0.7515   0.6331   0.5044   0.3770   0.2616   0.1662   0.0949   0.0766   0.0473   0.0197   0.0064   0.0024   0.0014   0.0001   0.0000

11
5    1.0000   0.9999   0.9986   0.9976   0.9936   0.9803   0.9527   0.9234   0.9051   0.8338   0.7384   0.6230   0.4956   0.3669   0.2485   0.2131   0.1503   0.0781   0.0328   0.0155   0.0099   0.0016   0.0001
6    1.0000   1.0000   0.9999   0.9997   0.9991   0.9965   0.9894   0.9803   0.9740   0.9452   0.8980   0.8281   0.7340   0.6177   0.4862   0.4407   0.3504   0.2241   0.1209   0.0697   0.0500   0.0128   0.0010
7    1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9984   0.9966   0.9952   0.9877   0.9726   0.9453   0.9004   0.8327   0.7384   0.7009   0.6172   0.4744   0.3222   0.2248   0.1798   0.0702   0.0115
8    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9995   0.9983   0.9955   0.9893   0.9767   0.9536   0.9140   0.8960   0.8507   0.7560   0.6242   0.5155   0.4557   0.2639   0.0861
9    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9990   0.9975   0.9940   0.9865   0.9827   0.9718   0.9437   0.8926   0.8385   0.8031   0.6513   0.4013
10    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

n = 12
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0     0.5404   0.2824   0.1422   0.1122   0.0687   0.0317   0.0138   0.0077   0.0057   0.0022   0.0008   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.8816   0.6590   0.4435   0.3813   0.2749   0.1584   0.0850   0.0540   0.0424   0.0196   0.0083   0.0032   0.0011   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.9804   0.8891   0.7358   0.6774   0.5583   0.3907   0.2528   0.1811   0.1513   0.0834   0.0421   0.0193   0.0079   0.0028   0.0008   0.0005   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
3    0.9978   0.9744   0.9078   0.8748   0.7946   0.6488   0.4925   0.3931   0.3467   0.2253   0.1345   0.0730   0.0356   0.0153   0.0056   0.0039   0.0017   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000
4    0.9998   0.9957   0.9761   0.9636   0.9274   0.8424   0.7237   0.6315   0.5833   0.4382   0.3044   0.1938   0.1117   0.0573   0.0255   0.0188   0.0095   0.0028   0.0006   0.0002   0.0001   0.0000   0.0000
5    1.0000   0.9995   0.9954   0.9921   0.9806   0.9456   0.8822   0.8223   0.7873   0.6652   0.5269   0.3872   0.2607   0.1582   0.0846   0.0664   0.0386   0.0143   0.0039   0.0013   0.0007   0.0001   0.0000
6    1.0000   0.9999   0.9993   0.9987   0.9961   0.9857   0.9614   0.9336   0.9154   0.8418   0.7393   0.6128   0.4731   0.3348   0.2127   0.1777   0.1178   0.0544   0.0194   0.0079   0.0046   0.0005   0.0000
7    1.0000   1.0000   0.9999   0.9998   0.9994   0.9972   0.9905   0.9812   0.9745   0.9427   0.8883   0.8062   0.6956   0.5618   0.4167   0.3685   0.2763   0.1576   0.0726   0.0364   0.0239   0.0043   0.0002
8    1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9983   0.9961   0.9944   0.9847   0.9644   0.9270   0.8655   0.7747   0.6533   0.6069   0.5075   0.3512   0.2054   0.1252   0.0922   0.0256   0.0022
9    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9995   0.9992   0.9972   0.9921   0.9807   0.9579   0.9166   0.8487   0.8189   0.7472   0.6093   0.4417   0.3226   0.2642   0.1109   0.0196
10    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9989   0.9968   0.9917   0.9804   0.9576   0.9460   0.9150   0.8416   0.7251   0.6187   0.5565   0.3410   0.1184
11    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9992   0.9978   0.9943   0.9923   0.9862   0.9683   0.9313   0.8878   0.8578   0.7176   0.4596
12    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000
CUMULATIVE BINOMIAL PROBABILITIES
n = 14
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0     0.4877   0.2288   0.1028   0.0779   0.0440   0.0178   0.0068   0.0034   0.0024   0.0008   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.8470   0.5846   0.3567   0.2960   0.1979   0.1010   0.0475   0.0274   0.0205   0.0081   0.0029   0.0009   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.9699   0.8416   0.6479   0.5795   0.4481   0.2811   0.1608   0.1053   0.0839   0.0398   0.0170   0.0065   0.0022   0.0006   0.0001   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
3    0.9958   0.9559   0.8535   0.8063   0.6982   0.5213   0.3552   0.2612   0.2205   0.1243   0.0632   0.0287   0.0114   0.0039   0.0011   0.0007   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
4    0.9996   0.9908   0.9533   0.9310   0.8702   0.7415   0.5842   0.4755   0.4227   0.2793   0.1672   0.0898   0.0426   0.0175   0.0060   0.0040   0.0017   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000
5    1.0000   0.9985   0.9885   0.9809   0.9561   0.8883   0.7805   0.6898   0.6405   0.4859   0.3373   0.2120   0.1189   0.0583   0.0243   0.0174   0.0083   0.0022   0.0004   0.0001   0.0000   0.0000   0.0000
6    1.0000   0.9998   0.9978   0.9959   0.9884   0.9617   0.9067   0.8505   0.8164   0.6925   0.5461   0.3953   0.2586   0.1501   0.0753   0.0576   0.0315   0.0103   0.0024   0.0007   0.0003   0.0000   0.0000
7    1.0000   1.0000   0.9997   0.9993   0.9976   0.9897   0.9685   0.9424   0.9247   0.8499   0.7414   0.6047   0.4539   0.3075   0.1836   0.1495   0.0933   0.0383   0.0116   0.0041   0.0022   0.0002   0.0000
8    1.0000   1.0000   1.0000   0.9999   0.9996   0.9978   0.9917   0.9826   0.9757   0.9417   0.8811   0.7880   0.6627   0.5141   0.3595   0.3102   0.2195   0.1117   0.0439   0.0191   0.0115   0.0015   0.0000
9    1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9983   0.9960   0.9940   0.9825   0.9574   0.9102   0.8328   0.7207   0.5773   0.5245   0.4158   0.2585   0.1298   0.0690   0.0467   0.0092   0.0004
10    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9993   0.9989   0.9961   0.9886   0.9713   0.9368   0.8757   0.7795   0.7388   0.6448   0.4787   0.3018   0.1937   0.1465   0.0441   0.0042
11    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9999   0.9994   0.9978   0.9935   0.9830   0.9602   0.9161   0.8947   0.8392   0.7189   0.5519   0.4205   0.3521   0.1584   0.0301
12    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9991   0.9971   0.9919   0.9795   0.9726   0.9525   0.8990   0.8021   0.7040   0.6433   0.4154   0.1530
13    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9992   0.9976   0.9966   0.9932   0.9822   0.9560   0.9221   0.8972   0.7712   0.5123
14    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

n = 16
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95

12
x=0     0.4401   0.1853   0.0743   0.0541   0.0281   0.0100   0.0033   0.0015   0.0010   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.8108   0.5147   0.2839   0.2272   0.1407   0.0635   0.0261   0.0137   0.0098   0.0033   0.0010   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.9571   0.7892   0.5614   0.4868   0.3518   0.1971   0.0994   0.0594   0.0451   0.0183   0.0066   0.0021   0.0006   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
3    0.9930   0.9316   0.7899   0.7291   0.5981   0.4050   0.2459   0.1659   0.1339   0.0651   0.0281   0.0106   0.0035   0.0009   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
4    0.9991   0.9830   0.9209   0.8866   0.7982   0.6302   0.4499   0.3391   0.2892   0.1666   0.0853   0.0384   0.0149   0.0049   0.0013   0.0008   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
5    0.9999   0.9967   0.9765   0.9622   0.9183   0.8103   0.6598   0.5469   0.4900   0.3288   0.1976   0.1051   0.0486   0.0191   0.0062   0.0040   0.0016   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000
6    1.0000   0.9995   0.9944   0.9899   0.9733   0.9204   0.8247   0.7374   0.6881   0.5272   0.3660   0.2272   0.1241   0.0583   0.0229   0.0159   0.0071   0.0016   0.0002   0.0000   0.0000   0.0000   0.0000
7    1.0000   0.9999   0.9989   0.9979   0.9930   0.9729   0.9256   0.8735   0.8406   0.7161   0.5629   0.4018   0.2559   0.1423   0.0671   0.0500   0.0257   0.0075   0.0015   0.0004   0.0002   0.0000   0.0000
8    1.0000   1.0000   0.9998   0.9996   0.9985   0.9925   0.9743   0.9500   0.9329   0.8577   0.7441   0.5982   0.4371   0.2839   0.1594   0.1265   0.0744   0.0271   0.0070   0.0021   0.0011   0.0001   0.0000
9    1.0000   1.0000   1.0000   1.0000   0.9998   0.9984   0.9929   0.9841   0.9771   0.9417   0.8759   0.7728   0.6340   0.4728   0.3119   0.2626   0.1753   0.0796   0.0267   0.0101   0.0056   0.0005   0.0000
10    1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9984   0.9960   0.9938   0.9809   0.9514   0.8949   0.8024   0.6712   0.5100   0.4531   0.3402   0.1897   0.0817   0.0378   0.0235   0.0033   0.0001
11    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9992   0.9987   0.9951   0.9851   0.9616   0.9147   0.8334   0.7108   0.6609   0.5501   0.3698   0.2018   0.1134   0.0791   0.0170   0.0009
12    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9991   0.9965   0.9894   0.9719   0.9349   0.8661   0.8341   0.7541   0.5950   0.4019   0.2709   0.2101   0.0684   0.0070
13    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9994   0.9979   0.9934   0.9817   0.9549   0.9406   0.9006   0.8029   0.6482   0.5132   0.4386   0.2108   0.0429
14    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9990   0.9967   0.9902   0.9863   0.9739   0.9365   0.8593   0.7728   0.7161   0.4853   0.1892
15    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9990   0.9985   0.9967   0.9900   0.9719   0.9459   0.9257   0.8147   0.5599
16    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000
CUMULATIVE BINOMIAL PROBABILITIES
n = 18
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0     0.3972   0.1501   0.0536   0.0376   0.0180   0.0056   0.0016   0.0007   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.7735   0.4503   0.2241   0.1728   0.0991   0.0395   0.0142   0.0068   0.0046   0.0013   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.9419   0.7338   0.4797   0.4027   0.2713   0.1353   0.0600   0.0326   0.0236   0.0082   0.0025   0.0007   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
3    0.9891   0.9018   0.7202   0.6479   0.5010   0.3057   0.1646   0.1017   0.0783   0.0328   0.0120   0.0038   0.0010   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
4    0.9985   0.9718   0.8794   0.8318   0.7164   0.5187   0.3327   0.2311   0.1886   0.0942   0.0411   0.0154   0.0049   0.0013   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
5    0.9998   0.9936   0.9581   0.9347   0.8671   0.7175   0.5344   0.4122   0.3550   0.2088   0.1077   0.0481   0.0183   0.0058   0.0014   0.0009   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
6    1.0000   0.9988   0.9882   0.9794   0.9487   0.8610   0.7217   0.6085   0.5491   0.3743   0.2258   0.1189   0.0537   0.0203   0.0062   0.0039   0.0014   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000
7    1.0000   0.9998   0.9973   0.9947   0.9837   0.9431   0.8593   0.7767   0.7283   0.5634   0.3915   0.2403   0.1280   0.0576   0.0212   0.0144   0.0061   0.0012   0.0002   0.0000   0.0000   0.0000   0.0000
8    1.0000   1.0000   0.9995   0.9989   0.9957   0.9807   0.9404   0.8924   0.8609   0.7368   0.5778   0.4073   0.2527   0.1347   0.0597   0.0433   0.0210   0.0054   0.0009   0.0002   0.0001   0.0000   0.0000
9    1.0000   1.0000   0.9999   0.9998   0.9991   0.9946   0.9790   0.9567   0.9403   0.8653   0.7473   0.5927   0.4222   0.2632   0.1391   0.1076   0.0596   0.0193   0.0043   0.0011   0.0005   0.0000   0.0000
10    1.0000   1.0000   1.0000   1.0000   0.9998   0.9988   0.9939   0.9856   0.9788   0.9424   0.8720   0.7597   0.6085   0.4366   0.2717   0.2233   0.1407   0.0569   0.0163   0.0053   0.0027   0.0002   0.0000
11    1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9986   0.9961   0.9938   0.9797   0.9463   0.8811   0.7742   0.6257   0.4509   0.3915   0.2783   0.1390   0.0513   0.0206   0.0118   0.0012   0.0000
12    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9991   0.9986   0.9942   0.9817   0.9519   0.8923   0.7912   0.6450   0.5878   0.4656   0.2825   0.1329   0.0653   0.0419   0.0064   0.0002
13    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9987   0.9951   0.9846   0.9589   0.9058   0.8114   0.7689   0.6673   0.4813   0.2836   0.1682   0.1206   0.0282   0.0015
14    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9990   0.9962   0.9880   0.9672   0.9217   0.8983   0.8354   0.6943   0.4990   0.3521   0.2798   0.0982   0.0109
15    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9993   0.9975   0.9918   0.9764   0.9674   0.9400   0.8647   0.7287   0.5973   0.5203   0.2662   0.0581
16    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9987   0.9954   0.9932   0.9858   0.9605   0.9009   0.8272   0.7759   0.5497   0.2265
17    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9993   0.9984   0.9944   0.9820   0.9624   0.9464   0.8499   0.6028
18    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

13
n = 20
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0     0.3585   0.1216   0.0388   0.0261   0.0115   0.0032   0.0008   0.0003   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.7358   0.3917   0.1756   0.1304   0.0692   0.0243   0.0076   0.0033   0.0021   0.0005   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.9245   0.6769   0.4049   0.3287   0.2061   0.0913   0.0355   0.0176   0.0121   0.0036   0.0009   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
3    0.9841   0.8670   0.6477   0.5665   0.4114   0.2252   0.1071   0.0604   0.0444   0.0160   0.0049   0.0013   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
4    0.9974   0.9568   0.8298   0.7687   0.6296   0.4148   0.2375   0.1515   0.1182   0.0510   0.0189   0.0059   0.0015   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
5    0.9997   0.9887   0.9327   0.8982   0.8042   0.6172   0.4164   0.2972   0.2454   0.1256   0.0553   0.0207   0.0064   0.0016   0.0003   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
6    1.0000   0.9976   0.9781   0.9629   0.9133   0.7858   0.6080   0.4793   0.4166   0.2500   0.1299   0.0577   0.0214   0.0065   0.0015   0.0009   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
7    1.0000   0.9996   0.9941   0.9887   0.9679   0.8982   0.7723   0.6615   0.6010   0.4159   0.2520   0.1316   0.0580   0.0210   0.0060   0.0037   0.0013   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000
8    1.0000   0.9999   0.9987   0.9972   0.9900   0.9591   0.8867   0.8095   0.7624   0.5956   0.4143   0.2517   0.1308   0.0565   0.0196   0.0130   0.0051   0.0009   0.0001   0.0000   0.0000   0.0000   0.0000
9    1.0000   1.0000   0.9998   0.9994   0.9974   0.9861   0.9520   0.9081   0.8782   0.7553   0.5914   0.4119   0.2493   0.1275   0.0532   0.0376   0.0171   0.0039   0.0006   0.0001   0.0000   0.0000   0.0000
10    1.0000   1.0000   1.0000   0.9999   0.9994   0.9961   0.9829   0.9624   0.9468   0.8725   0.7507   0.5881   0.4086   0.2447   0.1218   0.0919   0.0480   0.0139   0.0026   0.0006   0.0002   0.0000   0.0000
11    1.0000   1.0000   1.0000   1.0000   0.9999   0.9991   0.9949   0.9870   0.9804   0.9435   0.8692   0.7483   0.5857   0.4044   0.2376   0.1905   0.1133   0.0409   0.0100   0.0028   0.0013   0.0001   0.0000
12    1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9987   0.9963   0.9940   0.9790   0.9420   0.8684   0.7480   0.5841   0.3990   0.3385   0.2277   0.1018   0.0321   0.0113   0.0059   0.0004   0.0000
13    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9991   0.9985   0.9935   0.9786   0.9423   0.8701   0.7500   0.5834   0.5207   0.3920   0.2142   0.0867   0.0371   0.0219   0.0024   0.0000
14    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9997   0.9984   0.9936   0.9793   0.9447   0.8744   0.7546   0.7028   0.5836   0.3828   0.1958   0.1018   0.0673   0.0113   0.0003
15    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9985   0.9941   0.9811   0.9490   0.8818   0.8485   0.7625   0.5852   0.3704   0.2313   0.1702   0.0432   0.0026
16    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9987   0.9951   0.9840   0.9556   0.9396   0.8929   0.7748   0.5886   0.4335   0.3523   0.1330   0.0159
17    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9991   0.9964   0.9879   0.9824   0.9645   0.9087   0.7939   0.6713   0.5951   0.3231   0.0755
18    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9995   0.9979   0.9967   0.9924   0.9757   0.9308   0.8696   0.8244   0.6083   0.2642
19    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9997   0.9992   0.9968   0.9885   0.9739   0.9612   0.8784   0.6415
20    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000
CUMULATIVE BINOMIAL PROBABILITIES
n = 25
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0     0.2774   0.0718   0.0172   0.0105   0.0038   0.0008   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.6424   0.2712   0.0931   0.0629   0.0274   0.0070   0.0016   0.0005   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.8729   0.5371   0.2537   0.1887   0.0982   0.0321   0.0090   0.0035   0.0021   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
3    0.9659   0.7636   0.4711   0.3816   0.2340   0.0962   0.0332   0.0149   0.0097   0.0024   0.0005   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
4    0.9928   0.9020   0.6821   0.5937   0.4207   0.2137   0.0905   0.0462   0.0320   0.0095   0.0023   0.0005   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
5    0.9988   0.9666   0.8385   0.7720   0.6167   0.3783   0.1935   0.1120   0.0826   0.0294   0.0086   0.0020   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
6    0.9998   0.9905   0.9305   0.8908   0.7800   0.5611   0.3407   0.2215   0.1734   0.0736   0.0258   0.0073   0.0016   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
7    1.0000   0.9977   0.9745   0.9553   0.8909   0.7265   0.5118   0.3703   0.3061   0.1536   0.0639   0.0216   0.0058   0.0012   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
8    1.0000   0.9995   0.9920   0.9843   0.9532   0.8506   0.6769   0.5376   0.4668   0.2735   0.1340   0.0539   0.0174   0.0043   0.0008   0.0004   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
9    1.0000   0.9999   0.9979   0.9953   0.9827   0.9287   0.8106   0.6956   0.6303   0.4246   0.2424   0.1148   0.0440   0.0132   0.0029   0.0016   0.0005   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
10    1.0000   1.0000   0.9995   0.9988   0.9944   0.9703   0.9022   0.8220   0.7712   0.5858   0.3843   0.2122   0.0960   0.0344   0.0093   0.0056   0.0018   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000
11    1.0000   1.0000   0.9999   0.9997   0.9985   0.9893   0.9558   0.9082   0.8746   0.7323   0.5426   0.3450   0.1827   0.0778   0.0255   0.0164   0.0060   0.0009   0.0001   0.0000   0.0000   0.0000   0.0000
12    1.0000   1.0000   1.0000   0.9999   0.9996   0.9966   0.9825   0.9585   0.9396   0.8462   0.6937   0.5000   0.3063   0.1538   0.0604   0.0415   0.0175   0.0034   0.0004   0.0001   0.0000   0.0000   0.0000
13    1.0000   1.0000   1.0000   1.0000   0.9999   0.9991   0.9940   0.9836   0.9745   0.9222   0.8173   0.6550   0.4574   0.2677   0.1254   0.0918   0.0442   0.0107   0.0015   0.0003   0.0001   0.0000   0.0000
14    1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9982   0.9944   0.9907   0.9656   0.9040   0.7878   0.6157   0.4142   0.2288   0.1780   0.0978   0.0297   0.0056   0.0012   0.0005   0.0000   0.0000
15    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9995   0.9984   0.9971   0.9868   0.9560   0.8852   0.7576   0.5754   0.3697   0.3044   0.1894   0.0713   0.0173   0.0047   0.0021   0.0001   0.0000
16    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9992   0.9957   0.9826   0.9461   0.8660   0.7265   0.5332   0.4624   0.3231   0.1494   0.0468   0.0157   0.0080   0.0005   0.0000
17    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9988   0.9942   0.9784   0.9361   0.8464   0.6939   0.6297   0.4882   0.2735   0.1091   0.0447   0.0255   0.0023   0.0000

14
18    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9984   0.9927   0.9742   0.9264   0.8266   0.7785   0.6593   0.4389   0.2200   0.1092   0.0695   0.0095   0.0002
19    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9980   0.9914   0.9706   0.9174   0.8880   0.8065   0.6217   0.3833   0.2280   0.1615   0.0334   0.0012
20    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9995   0.9977   0.9905   0.9680   0.9538   0.9095   0.7863   0.5793   0.4063   0.3179   0.0980   0.0072
21    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9995   0.9976   0.9903   0.9851   0.9668   0.9038   0.7660   0.6184   0.5289   0.2364   0.0341
22    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9979   0.9965   0.9910   0.9679   0.9018   0.8113   0.7463   0.4629   0.1271
23    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9995   0.9984   0.9930   0.9726   0.9371   0.9069   0.7288   0.3576
24    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9992   0.9962   0.9895   0.9828   0.9282   0.7226
25    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000
CUMULATIVE BINOMIAL PROBABILITIES
n = 30
p       0.05      0.1     0.15      1/6      0.2     0.25      0.3      1/3     0.35      0.4     0.45      0.5     0.55      0.6     0.65      2/3      0.7     0.75      0.8      5/6     0.85      0.9     0.95
x=0     0.2146   0.0424   0.0076   0.0042   0.0012   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1    0.5535   0.1837   0.0480   0.0295   0.0105   0.0020   0.0003   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2    0.8122   0.4114   0.1514   0.1028   0.0442   0.0106   0.0021   0.0007   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
3    0.9392   0.6474   0.3217   0.2396   0.1227   0.0374   0.0093   0.0033   0.0019   0.0003   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
4    0.9844   0.8245   0.5245   0.4243   0.2552   0.0979   0.0302   0.0122   0.0075   0.0015   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
5    0.9967   0.9268   0.7106   0.6164   0.4275   0.2026   0.0766   0.0355   0.0233   0.0057   0.0011   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
6    0.9994   0.9742   0.8474   0.7765   0.6070   0.3481   0.1595   0.0838   0.0586   0.0172   0.0040   0.0007   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
7    0.9999   0.9922   0.9302   0.8863   0.7608   0.5143   0.2814   0.1668   0.1238   0.0435   0.0121   0.0026   0.0004   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
8    1.0000   0.9980   0.9722   0.9494   0.8713   0.6736   0.4315   0.2860   0.2247   0.0940   0.0312   0.0081   0.0016   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
9    1.0000   0.9995   0.9903   0.9803   0.9389   0.8034   0.5888   0.4317   0.3575   0.1763   0.0694   0.0214   0.0050   0.0009   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
10    1.0000   0.9999   0.9971   0.9933   0.9744   0.8943   0.7304   0.5848   0.5078   0.2915   0.1350   0.0494   0.0138   0.0029   0.0004   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
11    1.0000   1.0000   0.9992   0.9980   0.9905   0.9493   0.8407   0.7239   0.6548   0.4311   0.2327   0.1002   0.0334   0.0083   0.0014   0.0007   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
12    1.0000   1.0000   0.9998   0.9995   0.9969   0.9784   0.9155   0.8340   0.7802   0.5785   0.3592   0.1808   0.0714   0.0212   0.0045   0.0025   0.0006   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000
13    1.0000   1.0000   1.0000   0.9999   0.9991   0.9918   0.9599   0.9102   0.8737   0.7145   0.5025   0.2923   0.1356   0.0481   0.0124   0.0072   0.0021   0.0002   0.0000   0.0000   0.0000   0.0000   0.0000
14    1.0000   1.0000   1.0000   1.0000   0.9998   0.9973   0.9831   0.9565   0.9348   0.8246   0.6448   0.4278   0.2309   0.0971   0.0301   0.0188   0.0064   0.0008   0.0001   0.0000   0.0000   0.0000   0.0000
15    1.0000   1.0000   1.0000   1.0000   0.9999   0.9992   0.9936   0.9812   0.9699   0.9029   0.7691   0.5722   0.3552   0.1754   0.0652   0.0435   0.0169   0.0027   0.0002   0.0000   0.0000   0.0000   0.0000
16    1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9979   0.9928   0.9876   0.9519   0.8644   0.7077   0.4975   0.2855   0.1263   0.0898   0.0401   0.0082   0.0009   0.0001   0.0000   0.0000   0.0000
17    1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9994   0.9975   0.9955   0.9788   0.9286   0.8192   0.6408   0.4215   0.2198   0.1660   0.0845   0.0216   0.0031   0.0005   0.0002   0.0000   0.0000

15
18    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9993   0.9986   0.9917   0.9666   0.8998   0.7673   0.5689   0.3452   0.2761   0.1593   0.0507   0.0095   0.0020   0.0008   0.0000   0.0000
19    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9996   0.9971   0.9862   0.9506   0.8650   0.7085   0.4922   0.4152   0.2696   0.1057   0.0256   0.0067   0.0029   0.0001   0.0000
20    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9991   0.9950   0.9786   0.9306   0.8237   0.6425   0.5683   0.4112   0.1966   0.0611   0.0197   0.0097   0.0005   0.0000
21    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9984   0.9919   0.9688   0.9060   0.7753   0.7140   0.5685   0.3264   0.1287   0.0506   0.0278   0.0020   0.0000
22    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9996   0.9974   0.9879   0.9565   0.8762   0.8332   0.7186   0.4857   0.2392   0.1137   0.0698   0.0078   0.0001
23    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9993   0.9960   0.9828   0.9414   0.9162   0.8405   0.6519   0.3930   0.2235   0.1526   0.0258   0.0006
24    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9989   0.9943   0.9767   0.9645   0.9234   0.7974   0.5725   0.3836   0.2894   0.0732   0.0033
25    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9985   0.9925   0.9878   0.9698   0.9021   0.7448   0.5757   0.4755   0.1755   0.0156
26    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9981   0.9967   0.9907   0.9626   0.8773   0.7604   0.6783   0.3526   0.0608
27    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9997   0.9993   0.9979   0.9894   0.9558   0.8972   0.8486   0.5886   0.1878
28    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997   0.9980   0.9895   0.9705   0.9520   0.8163   0.4465
29    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9998   0.9988   0.9958   0.9924   0.9576   0.7854
30    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000
CUMULATIVE POISSON PROBABILITIES
λ               0.01      0.02     0.03     0.04     0.05     0.06     0.07     0.08     0.09
x=0             0.9900    0.9802   0.9704   0.9608   0.9512   0.9418   0.9324   0.9231   0.9139
1            1.0000    0.9998   0.9996   0.9992   0.9988   0.9983   0.9977   0.9970   0.9962
2            1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9999   0.9999
3            1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

λ               0.10      0.20     0.30     0.40     0.50     0.60     0.70     0.80     0.90
x=0             0.9048    0.8187   0.7408   0.6703   0.6065   0.5488   0.4966   0.4493   0.4066
1            0.9953    0.9825   0.9631   0.9384   0.9098   0.8781   0.8442   0.8088   0.7725
2            0.9998    0.9989   0.9964   0.9921   0.9856   0.9769   0.9659   0.9526   0.9371
3            1.0000    0.9999   0.9997   0.9992   0.9982   0.9966   0.9942   0.9909   0.9865
4            1.0000    1.0000   1.0000   0.9999   0.9998   0.9996   0.9992   0.9986   0.9977
5            1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9997
6            1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

λ      1.00     1.10      1.20     1.30     1.40     1.50     1.60     1.70     1.80     1.90
x=0    0.3679   0.3329    0.3012   0.2725   0.2466   0.2231   0.2019   0.1827   0.1653   0.1496
1   0.7358   0.6990    0.6626   0.6268   0.5918   0.5578   0.5249   0.4932   0.4628   0.4337
2   0.9197   0.9004    0.8795   0.8571   0.8335   0.8088   0.7834   0.7572   0.7306   0.7037
3   0.9810   0.9743    0.9662   0.9569   0.9463   0.9344   0.9212   0.9068   0.8913   0.8747
4   0.9963   0.9946    0.9923   0.9893   0.9857   0.9814   0.9763   0.9704   0.9636   0.9559
5   0.9994   0.9990    0.9985   0.9978   0.9968   0.9955   0.9940   0.9920   0.9896   0.9868
6   0.9999   0.9999    0.9997   0.9996   0.9994   0.9991   0.9987   0.9981   0.9974   0.9966
7   1.0000   1.0000    1.0000   0.9999   0.9999   0.9998   0.9997   0.9996   0.9994   0.9992
8   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9999   0.9998
9   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

λ      2.00     2.10      2.20     2.30     2.40     2.50     2.60     2.70     2.80     2.90
x=0    0.1353   0.1225    0.1108   0.1003   0.0907   0.0821   0.0743   0.0672   0.0608   0.0550
1   0.4060   0.3796    0.3546   0.3309   0.3084   0.2873   0.2674   0.2487   0.2311   0.2146
2   0.6767   0.6496    0.6227   0.5960   0.5697   0.5438   0.5184   0.4936   0.4695   0.4460
3   0.8571   0.8386    0.8194   0.7993   0.7787   0.7576   0.7360   0.7141   0.6919   0.6696
4   0.9473   0.9379    0.9275   0.9162   0.9041   0.8912   0.8774   0.8629   0.8477   0.8318
5   0.9834   0.9796    0.9751   0.9700   0.9643   0.9580   0.9510   0.9433   0.9349   0.9258
6   0.9955   0.9941    0.9925   0.9906   0.9884   0.9858   0.9828   0.9794   0.9756   0.9713
7   0.9989   0.9985    0.9980   0.9974   0.9967   0.9958   0.9947   0.9934   0.9919   0.9901
8   0.9998   0.9997    0.9995   0.9994   0.9991   0.9989   0.9985   0.9981   0.9976   0.9969
9   1.0000   0.9999    0.9999   0.9999   0.9998   0.9997   0.9996   0.9995   0.9993   0.9991
10   1.0000   1.0000    1.0000   1.0000   1.0000   0.9999   0.9999   0.9999   0.9998   0.9998
11   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999
12   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

λ      3.00     3.10      3.20     3.30     3.40     3.50     3.60     3.70     3.80     3.90
x=0    0.0498   0.0450    0.0408   0.0369   0.0334   0.0302   0.0273   0.0247   0.0224   0.0202
1   0.1991   0.1847    0.1712   0.1586   0.1468   0.1359   0.1257   0.1162   0.1074   0.0992
2   0.4232   0.4012    0.3799   0.3594   0.3397   0.3208   0.3027   0.2854   0.2689   0.2531
3   0.6472   0.6248    0.6025   0.5803   0.5584   0.5366   0.5152   0.4942   0.4735   0.4532
4   0.8153   0.7982    0.7806   0.7626   0.7442   0.7254   0.7064   0.6872   0.6678   0.6484
5   0.9161   0.9057    0.8946   0.8829   0.8705   0.8576   0.8441   0.8301   0.8156   0.8006
6   0.9665   0.9612    0.9554   0.9490   0.9421   0.9347   0.9267   0.9182   0.9091   0.8995
7   0.9881   0.9858    0.9832   0.9802   0.9769   0.9733   0.9692   0.9648   0.9599   0.9546
8   0.9962   0.9953    0.9943   0.9931   0.9917   0.9901   0.9883   0.9863   0.9840   0.9815
9   0.9989   0.9986    0.9982   0.9978   0.9973   0.9967   0.9960   0.9952   0.9942   0.9931
10   0.9997   0.9996    0.9995   0.9994   0.9992   0.9990   0.9987   0.9984   0.9981   0.9977
11   0.9999   0.9999    0.9999   0.9998   0.9998   0.9997   0.9996   0.9995   0.9994   0.9993
12   1.0000   1.0000    1.0000   1.0000   0.9999   0.9999   0.9999   0.9999   0.9998   0.9998
13   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999
14   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

16
CUMULATIVE POISSON PROBABILITIES
λ      4.00     4.10      4.20     4.30     4.40     4.50     4.60     4.70     4.80     4.90
x=0    0.0183   0.0166    0.0150   0.0136   0.0123   0.0111   0.0101   0.0091   0.0082   0.0074
1   0.0916   0.0845    0.0780   0.0719   0.0663   0.0611   0.0563   0.0518   0.0477   0.0439
2   0.2381   0.2238    0.2102   0.1974   0.1851   0.1736   0.1626   0.1523   0.1425   0.1333
3   0.4335   0.4142    0.3954   0.3772   0.3594   0.3423   0.3257   0.3097   0.2942   0.2793
4   0.6288   0.6093    0.5898   0.5704   0.5512   0.5321   0.5132   0.4946   0.4763   0.4582
5   0.7851   0.7693    0.7531   0.7367   0.7199   0.7029   0.6858   0.6684   0.6510   0.6335
6   0.8893   0.8786    0.8675   0.8558   0.8436   0.8311   0.8180   0.8046   0.7908   0.7767
7   0.9489   0.9427    0.9361   0.9290   0.9214   0.9134   0.9049   0.8960   0.8867   0.8769
8   0.9786   0.9755    0.9721   0.9683   0.9642   0.9597   0.9549   0.9497   0.9442   0.9382
9   0.9919   0.9905    0.9889   0.9871   0.9851   0.9829   0.9805   0.9778   0.9749   0.9717
10   0.9972   0.9966    0.9959   0.9952   0.9943   0.9933   0.9922   0.9910   0.9896   0.9880
11   0.9991   0.9989    0.9986   0.9983   0.9980   0.9976   0.9971   0.9966   0.9960   0.9953
12   0.9997   0.9997    0.9996   0.9995   0.9993   0.9992   0.9990   0.9988   0.9986   0.9983
13   0.9999   0.9999    0.9999   0.9998   0.9998   0.9997   0.9997   0.9996   0.9995   0.9994
14   1.0000   1.0000    1.0000   1.0000   0.9999   0.9999   0.9999   0.9999   0.9999   0.9998
15   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999
16   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

λ      5.00     5.50      6.00     6.50     7.00     7.50     8.00     8.50     9.00     9.50
x=0    0.0067   0.0041    0.0025   0.0015   0.0009   0.0006   0.0003   0.0002   0.0001   0.0001
1   0.0404   0.0266    0.0174   0.0113   0.0073   0.0047   0.0030   0.0019   0.0012   0.0008
2   0.1247   0.0884    0.0620   0.0430   0.0296   0.0203   0.0138   0.0093   0.0062   0.0042
3   0.2650   0.2017    0.1512   0.1118   0.0818   0.0591   0.0424   0.0301   0.0212   0.0149
4   0.4405   0.3575    0.2851   0.2237   0.1730   0.1321   0.0996   0.0744   0.0550   0.0403
5   0.6160   0.5289    0.4457   0.3690   0.3007   0.2414   0.1912   0.1496   0.1157   0.0885
6   0.7622   0.6860    0.6063   0.5265   0.4497   0.3782   0.3134   0.2562   0.2068   0.1649
7   0.8666   0.8095    0.7440   0.6728   0.5987   0.5246   0.4530   0.3856   0.3239   0.2687
8   0.9319   0.8944    0.8472   0.7916   0.7291   0.6620   0.5925   0.5231   0.4557   0.3918
9   0.9682   0.9462    0.9161   0.8774   0.8305   0.7764   0.7166   0.6530   0.5874   0.5218
10   0.9863   0.9747    0.9574   0.9332   0.9015   0.8622   0.8159   0.7634   0.7060   0.6453
11   0.9945   0.9890    0.9799   0.9661   0.9467   0.9208   0.8881   0.8487   0.8030   0.7520
12   0.9980   0.9955    0.9912   0.9840   0.9730   0.9573   0.9362   0.9091   0.8758   0.8364
13   0.9993   0.9983    0.9964   0.9929   0.9872   0.9784   0.9658   0.9486   0.9261   0.8981
14   0.9998   0.9994    0.9986   0.9970   0.9943   0.9897   0.9827   0.9726   0.9585   0.9400
15   0.9999   0.9998    0.9995   0.9988   0.9976   0.9954   0.9918   0.9862   0.9780   0.9665
16   1.0000   0.9999    0.9998   0.9996   0.9990   0.9980   0.9963   0.9934   0.9889   0.9823
17   1.0000   1.0000    0.9999   0.9998   0.9996   0.9992   0.9984   0.9970   0.9947   0.9911
18   1.0000   1.0000    1.0000   0.9999   0.9999   0.9997   0.9993   0.9987   0.9976   0.9957
19   1.0000   1.0000    1.0000   1.0000   1.0000   0.9999   0.9997   0.9995   0.9989   0.9980
20   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9996   0.9991
21   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9996
22   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9999
23   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999
24   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

17
CUMULATIVE POISSON PROBABILITIES
λ     10.00    11.00     12.00    13.00    14.00    15.00    16.00    17.00    18.00    19.00
x=0    0.0000   0.0000    0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
1   0.0005   0.0002    0.0001   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
2   0.0028   0.0012    0.0005   0.0002   0.0001   0.0000   0.0000   0.0000   0.0000   0.0000
3   0.0103   0.0049    0.0023   0.0011   0.0005   0.0002   0.0001   0.0000   0.0000   0.0000
4   0.0293   0.0151    0.0076   0.0037   0.0018   0.0009   0.0004   0.0002   0.0001   0.0000
5   0.0671   0.0375    0.0203   0.0107   0.0055   0.0028   0.0014   0.0007   0.0003   0.0002
6   0.1301   0.0786    0.0458   0.0259   0.0142   0.0076   0.0040   0.0021   0.0010   0.0005
7   0.2202   0.1432    0.0895   0.0540   0.0316   0.0180   0.0100   0.0054   0.0029   0.0015
8   0.3328   0.2320    0.1550   0.0998   0.0621   0.0374   0.0220   0.0126   0.0071   0.0039
9   0.4579   0.3405    0.2424   0.1658   0.1094   0.0699   0.0433   0.0261   0.0154   0.0089
10   0.5830   0.4599    0.3472   0.2517   0.1757   0.1185   0.0774   0.0491   0.0304   0.0183
11   0.6968   0.5793    0.4616   0.3532   0.2600   0.1848   0.1270   0.0847   0.0549   0.0347
12   0.7916   0.6887    0.5760   0.4631   0.3585   0.2676   0.1931   0.1350   0.0917   0.0606
13   0.8645   0.7813    0.6815   0.5730   0.4644   0.3632   0.2745   0.2009   0.1426   0.0984
14   0.9165   0.8540    0.7720   0.6751   0.5704   0.4657   0.3675   0.2808   0.2081   0.1497
15   0.9513   0.9074    0.8444   0.7636   0.6694   0.5681   0.4667   0.3715   0.2867   0.2148
16   0.9730   0.9441    0.8987   0.8355   0.7559   0.6641   0.5660   0.4677   0.3751   0.2920
17   0.9857   0.9678    0.9370   0.8905   0.8272   0.7489   0.6593   0.5640   0.4686   0.3784
18   0.9928   0.9823    0.9626   0.9302   0.8826   0.8195   0.7423   0.6550   0.5622   0.4695
19   0.9965   0.9907    0.9787   0.9573   0.9235   0.8752   0.8122   0.7363   0.6509   0.5606
20   0.9984   0.9953    0.9884   0.9750   0.9521   0.9170   0.8682   0.8055   0.7307   0.6472
21   0.9993   0.9977    0.9939   0.9859   0.9712   0.9469   0.9108   0.8615   0.7991   0.7255
22   0.9997   0.9990    0.9970   0.9924   0.9833   0.9673   0.9418   0.9047   0.8551   0.7931
23   0.9999   0.9995    0.9985   0.9960   0.9907   0.9805   0.9633   0.9367   0.8989   0.8490
24   1.0000   0.9998    0.9993   0.9980   0.9950   0.9888   0.9777   0.9594   0.9317   0.8933
25   1.0000   0.9999    0.9997   0.9990   0.9974   0.9938   0.9869   0.9748   0.9554   0.9269
26   1.0000   1.0000    0.9999   0.9995   0.9987   0.9967   0.9925   0.9848   0.9718   0.9514
27   1.0000   1.0000    0.9999   0.9998   0.9994   0.9983   0.9959   0.9912   0.9827   0.9687
28   1.0000   1.0000    1.0000   0.9999   0.9997   0.9991   0.9978   0.9950   0.9897   0.9805
29   1.0000   1.0000    1.0000   1.0000   0.9999   0.9996   0.9989   0.9973   0.9941   0.9882
30   1.0000   1.0000    1.0000   1.0000   0.9999   0.9998   0.9994   0.9986   0.9967   0.9930
31   1.0000   1.0000    1.0000   1.0000   1.0000   0.9999   0.9997   0.9993   0.9982   0.9960
32   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   0.9999   0.9996   0.9990   0.9978
33   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9995   0.9988
34   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998   0.9994
35   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9997
36   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999   0.9998
37   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   0.9999
38   1.0000   1.0000    1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

18
THE NORMAL DISTRIBUTION FUNCTION

If Z has a normal distribution with mean 0 and
variance 1 then, for each value of , the table gives
the value of Φ( ), where
Φ( ) = P(Z ≤ ).

For negative values of          use Φ(− ) = 1 − Φ( ).

0        1         2        3         4        5        6           7     8        9      1 2 3 4 5 6 7 8 9
0.0   0.5000   0.5040   0.5080    0.5120    0.5160   0.5199   0.5239   0.5279   0.5319   0.5359   4   8   12   16   20   24   28   32   36
0.1   0.5398   0.5438   0.5478    0.5517    0.5557   0.5596   0.5636   0.5675   0.5714   0.5753   4   8   12   16   20   24   28   32   36
0.2   0.5793   0.5832   0.5871    0.5910    0.5948   0.5987   0.6026   0.6064   0.6103   0.6141   4   8   12   15   19   23   27   31   35
0.3   0.6179   0.6217   0.6255    0.6293    0.6331   0.6368   0.6406   0.6443   0.6480   0.6517   4   7   11   15   19   22   26   30   34
0.4   0.6554   0.6591   0.6628    0.6664    0.6700   0.6736   0.6772   0.6808   0.6844   0.6879   4   7   11   14   18   22   25   29   32
0.5   0.6915   0.6950   0.6985    0.7019    0.7054   0.7088   0.7123   0.7157   0.7190   0.7224   3   7 10 14 17 20 24             27   31
0.6   0.7257   0.7291   0.7324    0.7357    0.7389   0.7422   0.7454   0.7486   0.7517   0.7549   3   7 10 13 16 19 23             26   29
0.7   0.7580   0.7611   0.7642    0.7673    0.7704   0.7734   0.7764   0.7794   0.7823   0.7852   3   6 9 12 15 18 21              24   27
0.8   0.7881   0.7910   0.7939    0.7967    0.7995   0.8023   0.8051   0.8078   0.8106   0.8133   3   5 8 11 14 16 19              22   25
0.9   0.8159   0.8186   0.8212    0.8238    0.8264   0.8289   0.8315   0.8340   0.8365   0.8389   3   5 8 10 13 15 18              20   23
1.0   0.8413   0.8438   0.8461    0.8485    0.8508   0.8531   0.8554   0.8577   0.8599   0.8621   2   5    7    9 12 14 16 19 21
1.1   0.8643   0.8665   0.8686    0.8708    0.8729   0.8749   0.8770   0.8790   0.8810   0.8830   2   4    6    8 10 12 14 16 18
1.2   0.8849   0.8869   0.8888    0.8907    0.8925   0.8944   0.8962   0.8980   0.8997   0.9015   2   4    6    7 9 11 13 15 17
1.3   0.9032   0.9049   0.9066    0.9082    0.9099   0.9115   0.9131   0.9147   0.9162   0.9177   2   3    5    6 8 10 11 13 14
1.4   0.9192   0.9207   0.9222    0.9236    0.9251   0.9265   0.9279   0.9292   0.9306   0.9319   1   3    4    6 7 8 10 11 13
1.5   0.9332   0.9345   0.9357    0.9370    0.9382   0.9394   0.9406   0.9418   0.9429   0.9441   1   2    4    5    6    7    8 10 11
1.6   0.9452   0.9463   0.9474    0.9484    0.9495   0.9505   0.9515   0.9525   0.9535   0.9545   1   2    3    4    5    6    7 8 9
1.7   0.9554   0.9564   0.9573    0.9582    0.9591   0.9599   0.9608   0.9616   0.9625   0.9633   1   2    3    4    4    5    6 7 8
1.8   0.9641   0.9649   0.9656    0.9664    0.9671   0.9678   0.9686   0.9693   0.9699   0.9706   1   1    2    3    4    4    5 6 6
1.9   0.9713   0.9719   0.9726    0.9732    0.9738   0.9744   0.9750   0.9756   0.9761   0.9767   1   1    2    2    3    4    4 5 5
2.0   0.9772   0.9778   0.9783    0.9788    0.9793   0.9798   0.9803   0.9808   0.9812   0.9817   0   1    1    2    2    3    3    4    4
2.1   0.9821   0.9826   0.9830    0.9834    0.9838   0.9842   0.9846   0.9850   0.9854   0.9857   0   1    1    2    2    2    3    3    4
2.2   0.9861   0.9864   0.9868    0.9871    0.9875   0.9878   0.9881   0.9884   0.9887   0.9890   0   1    1    1    2    2    2    3    3
2.3   0.9893   0.9896   0.9898    0.9901    0.9904   0.9906   0.9909   0.9911   0.9913   0.9916   0   1    1    1    1    2    2    2    2
2.4   0.9918   0.9920   0.9922    0.9925    0.9927   0.9929   0.9931   0.9932   0.9934   0.9936   0   0    1    1    1    1    1    2    2
2.5   0.9938   0.9940   0.9941    0.9943    0.9945   0.9946   0.9948   0.9949   0.9951   0.9952   0   0    0    1    1    1    1    1    1
2.6   0.9953   0.9955   0.9956    0.9957    0.9959   0.9960   0.9961   0.9962   0.9963   0.9964   0   0    0    0    1    1    1    1    1
2.7   0.9965   0.9966   0.9967    0.9968    0.9969   0.9970   0.9971   0.9972   0.9973   0.9974   0   0    0    0    0    1    1    1    1
2.8   0.9974   0.9975   0.9976    0.9977    0.9977   0.9978   0.9979   0.9979   0.9980   0.9981   0   0    0    0    0    0    0    1    1
2.9   0.9981   0.9982   0.9982    0.9983    0.9984   0.9984   0.9985   0.9985   0.9986   0.9986   0   0    0    0    0    0    0    0    0

Critical values for the normal distribution
If Z has a normal distribution with mean 0 and
variance 1 then, for each value of p, the table gives
the value of such that
P(Z ≤ ) = p.

p       0.75           0.90         0.95            0.975          0.99        0.995          0.9975          0.999          0.9995

0.674          1.282        1.645           1.960          2.326       2.576          2.807           3.090          3.291

19
CRITICAL VALUES FOR THE t DISTRIBUTION

If T has a t distribution with v degrees of freedom
then, for each pair of values of p and v, the table
gives the value of t such that
P(T ≤ t) = p.

p       0.75    0.90    0.95     0.975      0.99    0.995    0.9975    0.999     0.9995
v=1       1.000   3.078   6.314    12.71     31.82    63.66    127.3     318.3     636.6
2       0.816   1.886   2.920     4.303     6.965    9.925    14.09     22.33     31.60
3       0.765   1.638   2.353     3.182     4.541    5.841     7.453    10.21     12.92
4       0.741   1.533   2.132     2.776     3.747    4.604     5.598     7.173     8.610
5   0.727   1.476   2.015     2.571     3.365    4.032     4.773     5.894     6.869
6   0.718   1.440   1.943     2.447     3.143    3.707     4.317     5.208     5.959
7   0.711   1.415   1.895     2.365     2.998    3.499     4.029     4.785     5.408
8   0.706   1.397   1.860     2.306     2.896    3.355     3.833     4.501     5.041
9   0.703   1.383   1.833     2.262     2.821    3.250     3.690     4.297     4.781
10     0.700   1.372   1.812     2.228     2.764    3.169     3.581     4.144     4.587
11     0.697   1.363   1.796     2.201     2.718    3.106     3.497     4.025     4.437
12     0.695   1.356   1.782     2.179     2.681    3.055     3.428     3.930     4.318
13     0.694   1.350   1.771     2.160     2.650    3.012     3.372     3.852     4.221
14     0.692   1.345   1.761     2.145     2.624    2.977     3.326     3.787     4.140
15     0.691   1.341   1.753     2.131     2.602    2.947     3.286     3.733     4.073
16     0.690   1.337   1.746     2.120     2.583    2.921     3.252     3.686     4.015
17     0.689   1.333   1.740     2.110     2.567    2.898     3.222     3.646     3.965
18     0.688   1.330   1.734     2.101     2.552    2.878     3.197     3.610     3.922
19     0.688   1.328   1.729     2.093     2.539    2.861     3.174     3.579     3.883
20     0.687   1.325   1.725     2.086     2.528    2.845     3.153     3.552     3.850
21     0.686   1.323   1.721     2.080     2.518    2.831     3.135     3.527     3.819
22     0.686   1.321   1.717     2.074     2.508    2.819     3.119     3.505     3.792
23     0.685   1.319   1.714     2.069     2.500    2.807     3.104     3.485     3.768
24     0.685   1.318   1.711     2.064     2.492    2.797     3.091     3.467     3.745
25     0.684   1.316   1.708     2.060     2.485    2.787     3.078     3.450     3.725
26     0.684   1.315   1.706     2.056     2.479    2.779     3.067     3.435     3.707
27     0.684   1.314   1.703     2.052     2.473    2.771     3.057     3.421     3.689
28     0.683   1.313   1.701     2.048     2.467    2.763     3.047     3.408     3.674
29     0.683   1.311   1.699     2.045     2.462    2.756     3.038     3.396     3.660
30     0.683   1.310   1.697     2.042     2.457    2.750     3.030     3.385     3.646
40     0.681   1.303   1.684     2.021     2.423    2.704     2.971     3.307     3.551
60     0.679   1.296   1.671     2.000     2.390    2.660     2.915     3.232     3.460
120     0.677   1.289   1.658     1.980     2.358    2.617     2.860     3.160     3.373
∞      0.674   1.282   1.645     1.960     2.326    2.576     2.807     3.090     3.291

20
CRITICAL VALUES FOR THE χ2 DISTRIBUTION

If X has a χ 2 distribution with v degrees of freedom
then, for each pair of values of p and v, the table
gives the value of x such that
P(X ≤ x) = p.

p         0.01          0.025         0.05        0.90      0.95     0.975      0.99     0.995     0.999
v=1        0.03 1571     0.03 9821     0.02 3932     2.706     3.841     5.024     6.635     7.879    10.83
2        0.02010       0.05064       0.1026        4.605     5.991     7.378     9.210    10.60     13.82
3        0.1148        0.2158        0.3518        6.251     7.815     9.348    11.34     12.84     16.27
4        0.2971        0.4844        0.7107        7.779     9.488    11.14     13.28     14.86     18.47
5    0.5543        0.8312        1.145         9.236    11.07     12.83     15.09     16.75     20.51
6    0.8721        1.237         1.635        10.64     12.59     14.45     16.81     18.55     22.46
7    1.239         1.690         2.167        12.02     14.07     16.01     18.48     20.28     24.32
8    1.647         2.180         2.733        13.36     15.51     17.53     20.09     21.95     26.12
9    2.088         2.700         3.325        14.68     16.92     19.02     21.67     23.59     27.88
10       2.558         3.247         3.940        15.99     18.31     20.48     23.21     25.19     29.59
11       3.053         3.816         4.575        17.28     19.68     21.92     24.73     26.76     31.26
12       3.571         4.404         5.226        18.55     21.03     23.34     26.22     28.30     32.91
13       4.107         5.009         5.892        19.81     22.36     24.74     27.69     29.82     34.53
14       4.660         5.629         6.571        21.06     23.68     26.12     29.14     31.32     36.12
15       5.229         6.262        7.261         22.31     25.00     27.49     30.58     32.80     37.70
16       5.812         6.908        7.962         23.54     26.30     28.85     32.00     34.27     39.25
17       6.408         7.564        8.672         24.77     27.59     30.19     33.41     35.72     40.79
18       7.015         8.231        9.390         25.99     28.87     31.53     34.81     37.16     42.31
19       7.633         8.907       10.12          27.20     30.14     32.85     36.19     38.58     43.82
20       8.260         9.591       10.85          28.41     31.41     34.17     37.57     40.00     45.31
21       8.897        10.28        11.59          29.62     32.67     35.48     38.93     41.40     46.80
22       9.542        10.98        12.34          30.81     33.92     36.78     40.29     42.80     48.27
23      10.20         11.69        13.09          32.01     35.17     38.08     41.64     44.18     49.73
24      10.86         12.40        13.85          33.20     36.42     39.36     42.98     45.56     51.18
25      11.52         13.12        14.61          34.38     37.65     40.65     44.31     46.93     52.62
30      14.95         16.79        18.49          40.26     43.77     46.98     50.89     53.67     59.70
40      22.16         24.43        26.51          51.81     55.76     59.34     63.69     66.77     73.40
50      29.71         32.36        34.76          63.17     67.50     71.42     76.15     79.49     86.66
60      37.48         40.48        43.19          74.40     79.08     83.30     88.38     91.95     99.61
70      45.44         48.76        51.74          85.53     90.53     95.02    100.4     104.2     112.3
80      53.54         57.15        60.39          96.58    101.9     106.6     112.3     116.3     124.8
90      61.75         65.65        69.13         107.6     113.1     118.1     124.1     128.3     137.2
100      70.06         74.22        77.93         118.5     124.3     129.6     135.8     140.2     149.4

21
WILCOXON SIGNED RANK TEST

P is the sum of the ranks corresponding to the positive differences,
Q is the sum of the ranks corresponding to the negative differences,
T is the smaller of P and Q.

For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at the level
of signiﬁcance indicated.

Critical values of T

Level of signiﬁcance
One Tail       0.05      0.025      0.01      0.005
Two Tail       0.10      0.05       0.02      0.01
n=6            2          0
7           3          2          0
8           5          3          1         0
9           8          5          3         1
10          10          8          5         3
11          13         10          7         5
12          17         13          9         7
13          21         17         12         9
14          25         21         15        12
15          30         25         19        15
16          35         29         23        19
17          41         34         27        23
18          47         40         32        27
19          53         46         37        32
20          60         52         43        37

For larger values of n, each of P and Q can be approximated by the normal distribution with mean 1 n(n + 1) and
4
variance 24 n(n + 1)(2n + 1).
1

22
WILCOXON RANK SUM TEST

The two samples have sizes m and n, where m ≤ n.
Rm is the sum of the ranks of the items in the sample of size m.
W is the smaller of Rm and m(m + n + 1) − Rm .

For each pair of values of m and n, the table gives the largest value of W which will lead to rejection of the null
hypothesis at the level of signiﬁcance indicated.

Critical values of W

Level of signiﬁcance
One Tail    0.05   0.025    0.01    0.05    0.025      0.01   0.05   0.025     0.01   0.05   0.025      0.01
Two Tail     0.1    0.05    0.02    0.1      0.05      0.02   0.1    0.05      0.02   0.1    0.05       0.02
n              m=3                      m=4                      m=5                     m=6
3         6       −       −
4         6       −       −      11       10        −
5         7       6       −      12       11        10     19         17   16
6         8       7       −      13       12        11     20         18   17     28       26        24
7         8       7       6      14       13        11     21         20   18     29       27        25
8         9       8       6      15       14        12     23         21   19     31       29        27
9        10       8       7      16       14        13     24         22   20     33       31        28
10        10       9       7      17       15        13     26         23   21     35       32        29

Level of signiﬁcance
One Tail    0.05   0.025    0.01    0.05    0.025      0.01   0.05   0.025     0.01   0.05   0.025      0.01
Two Tail     0.1    0.05    0.02    0.1      0.05      0.02   0.1    0.05      0.02   0.1    0.05       0.02
n              m=7                      m=8                      m=9                     m = 10
7        39      36      34
8        41      38      35      51       49        45
9        43      40      37      54       51        47     66         62   59
10        45      42      39      56       53        49     69         65   61     82       78        74

For larger values of m and n, the normal distribution with mean 1 m(m + n + 1) and variance
2
1
12
mn(m     + n + 1) should
be used as an approximation to the distribution of Rm .

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